Domain and range/codomain

A function with variables, has a subset of as domain and a subset of as codomain.

Derivative

The derivative remains the same, but we will need to take partial derivatives with respect to each single variable.

Example

makes a 3D paraboloid.

  • Curves
  • Ling integrals
  • Double and triple integrals
  • Vector fields
  • Surfaces
Scalar function?
Example

The domain is
Which leads to , which is a circle. So the domain in this case are all the values inside the circle.

  1. Interior point: A point belongs to the domain () and is an interior point if and only if there exists an such that:

This means that there is an open disk centered at with radius that is entirely contained within the domain .

  1. Boundary point: A point belongs to the domain () and is a boundary point if and only if for every :

and

Danger

This is completely reconstructed, it is probably inaccurate.

Example

Domain

, or

Danger

Recap on how to compute domains of everything is needed.

Example

Domain

Example

Domain

Example

Domain

  • (circle centered at 0,0 with radius )
    • (ellipse)

Danger

You cannot just square both sides of an inequality LOL

Example

Domain

For the first one:


  • or


  • Manca un'immagine

Limits for function of two variables

Limit exists when both left limit and right limit exist. So if from different directions the limit approaches the same value.

Example



wip

(the easiest thing to try)

The limit does not exist, since it depends on the particular direction we are considering.